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  • Simple computer code for estimating cosmic-ray shieldingby oddly shaped objects

    Greg Balcoa,

    a Berkeley Geochronology Center, 2455 Ridge Road, Berkeley CA 94709 USA


    This paper describes computer code that estimates the effect on cosmogenic-nuclideproduction rates of arbitrarily shaped obstructions that are partially or completely opaqueto cosmic rays. This is potentially useful for cosmogenic-nuclide exposure dating ofgeometrically complex landforms. The code has been validated against analytical for-mulae applicable to objects with regular geometries. It has not yet been validatedagainst empirical measurements of cosmogenic-nuclide concentrations in samples withthe same exposure history but different shielding geometries.

    Keywords: Cosmogenic nuclide geochemistry, Exposure-age dating, Monte Carlointegration, Precariously balanced rocks

    1. Importance of geometric shielding of the cosmic-ray flux to exposure dating 1

    Cosmogenic-nuclide exposure dating is a geochemical method used to determine 2the age of geological events that create or modify the Earths surface, such as glacier 3advances and retreats, landslides, earthquake surface ruptures, or instances of erosion 4or sediment deposition. To determine the exposure age of a rock surface, one must 5i) measure the concentration of a trace cosmic-ray-produced nuclide (e.g., 10Be, 26Al, 63He, etc.); and ii) use an independently calibrated nuclide production rate to interpret 7the concentration as an exposure age (see review in Dunai, 2010). 8

    Typically (Dunai, 2010), estimating the production rate at a sample site employs 9simplifying assumptions that i) the sample is located on an infinite flat surface, and ii) 10any topographic obstructions to the cosmic-ray flux can be represented as an apparent 11horizon below which cosmic rays are fully obstructed, and above which they are fully 12admitted. These assumptions are adequate for a very wide range of useful exposure- 13dating applications, but they fail when the sample site is located on or near objects 14whose dimensions are similar to the cosmic ray particle attenuation length in rock 15(order 1 meter). 16

    The main reason this assumption fails is referred to in this paper as geometric 17shielding. Geometric shielding describes the effect that when a sample is surrounded 18

    Corresponding author. Tel. 510.644.9200 Fax 510.644.9201Email address: [email protected] (Greg Balco )

    Preprint submitted to Quaternary Geochronology October 3, 2013

  • by obstructions, the cosmic-ray flux from certain directions must pass through rock 19or sediment, rather than only air, before reaching the sample site. If the thickness of 20rock traversed is long relative to the cosmic-ray attenuation length, then the cosmic-ray 21flux from that direction is completely attenuated and the opaque-horizon assumption 22described above is adequate. However, if it is relatively short then the cosmic-ray 23flux is only partially attenuated and this assumption fails. The term self-shielding 24is sometimes used to describe this effect, although this term is somewhat misleading 25because both the rock that is being sampled (self) and other nearby objects could both 26shield the sample site. 27

    Another effect that is potentially important for samples not embedded in infinite flat 28surfaces is here called the missing mass effect; Masarik and Wieler (2003) also called 29this the shape effect. This describes the fact that a minority of cosmogenic-nuclide 30production at or beneath the Earths surface is due to secondary particles originating 31from nuclear reactions within the surrounding rock mass. Given a sample embedded 32in an infinite flat surface, secondary particles responsible for nuclide production in the 33sample are produced all around the sample site. However, when a sample location is 34in part surrounded by air rather than rock for example if it lies on a steeply dipping 35surface, or within a relatively small boulder the flux of these secondary particles, and 36therefore the production rate, is less. 37

    The magnitude of geometric shielding can vary from negligible to complete atten- 38uation depending on the arrangement and size of obstructions. The magnitude of the 39missing mass effect is expected to be at most 10% of the production rate (i.e., for 40the extreme case in which a sample lies at the top of a tall, thin pillar; Masarik and 41Wieler, 2003). Thus, in many common geomorphic situations, geometric shielding is 42significantly more important (a tens-of-percent-level effect on the production rate) than 43the missing mass effect (a percent-level effect). Geometric shielding is also easier to 44calculate. The remainder of this paper describes computer code that computes geo- 45metric shielding for arbitrary geometry of samples and obstructions; this computation 46only requires computing the lengths of ray paths that pass through objects and apply- 47ing an exponential attenuation factor to each ray path. Computer code to implement 48this is simple and fast enough to permit, for example, dynamic calculation of shielding 49factors in a model where the shielding geometry evolves during the exposure history 50of a sample. Previous attempts to quantify the missing mass effect, on the other hand, 51employed a complete simulation of the cascade of nuclear reactions in the atmosphere 52and rock that give rise to the particle flux at the sample site, using first-principles par- 53ticle interaction codes such as MCNP or GEANT (Masarik and Beer, 1999, and ref- 54erences therein). These codes are slow, complex, and require extensive computational 55resources. To summarize, because of the difference in the potential magnitude of these 56two effects, there are many geomorphically relevant situations involving obstructed 57samples in which one can estimate production rates at useful (i.e., percent-level) ac- 58curacy using only the relatively simple geometric shielding calculation. The calcula- 59tions in this paper consider only the geometric shielding effect and do not consider the 60missing mass effect. A valuable future contribution would be to develop a simplified 61method of approximating the missing mass effect using only geometric considerations 62that would not require complex particle physics simulation code. 63

    Although both the geometric shielding effect and the missing mass effect follow 64


  • from well-established physical principles, there has been very little attempt to verify 65calculations of these effects by measuring cosmogenic-nuclide concentrations in natu- 66ral situations where they are expected to be important. For example, the missing mass 67effect predicts a relationship between boulder size, sample location on a boulder, and 68cosmogenic-nuclide concentration. Although some researchers have argued that this 69relationship is or is not present in certain exposure-age data sets (Masarik and Wieler, 702003; Balco and Schaefer, 2006), the scatter in these data sets is similar in magnitude 71to expected effects, so these results were ambiguous. A more comprehensive example 72is the study of Kubik and Reuther (2007), who attempted to match cosmogenic 10Be 73concentrations measured on and beneath a cliff surface, where both geometric shield- 74ing and missing mass effects would be expected to be important, with model estimates 75of these effects. However, they were not able to reconcile observations with predictions 76at high confidence. 77

    To summarize, a method of computing cosmogenic-nuclide production rates on, 78near, and within arbitrarily-shaped objects is potentially useful for exposure-dating ap- 79plications, because many landforms of geomorphic and geologic interest have complex 80shapes at the scale of cosmic-ray attenuation lengths. As discussed above and as noted 81by others including Dunne et al. (1999), Lal and Chen (2005), and Mackey and Lamb 82(2013), in many cases this can be accomplished at adequate precision by considering 83only geometric shielding. The computer code described in this paper was originally 84developed for exposure-dating of precariously balanced rocks used in seismic hazard 85estimation (Balco et al., 2011). These rocks are gradually exhumed by erosion of sur- 86rounding soil, and determining the rate and timing of this exhumation requires collect- 87ing samples on the sides of the rocks at a range of heights. As most of these rocks are 881-2 meters in size and irregular in shape (e.g., Figure 1), these sample locations are 89subject to partial shielding by the rock itself, and the magnitude of this shielding varies 90with sample location. In addition, when the rock is partially exhumed, sample sites can 91also be shielded by soil. To calculate these effects, in Balco et al. (2011) we developed 92computer code to calculate geometric shielding factors for arbitrarily shaped objects. 93However, that paper did not include the code or describe it in detail. The present paper 94provides a complete description and includes the code as supplementary information. 95

    For purposes of review of this paper, the MATLAB code, documentation, and 96ancillary information is available online at: 97 98


    2. Definition of geometric shielding; previous work 100

    The treatment of geometric shielding in this paper follows that of Lal (1991), which 101was also adopted by Dunne et al. (1999), Lal and Chen (2005), and Mackey and Lamb 102(2013). Dunne et al. (1999) provides a complete and clear derivation of the following 103equations. Cosmic ray intensity as a function of direction I(, ), where is the zenith 104angle measured down from the vertical and is the azimuthal angle, is assumed to be 105constant in azimuth but vary with zenith angle, such that: 106


  • I(, ) = I0 cosm